This problems is from Jack Lee's Introduction to topological manifold, problem 7-15.
Suppose you have 3 circles sitting in $\mathbb{R}^2$ along the x axis and are tangent to neighboring circles (so the middle circle is tangent to the circle on the left and right), prove that it is homotopic equivalent to the bouquet of 3 circles.
I am aware of the "standard answer" using the hint given by the problem, by showing that both spaces are strong deformation retract of $\mathbb{R}^2$ with 3 holes. However, is there a more explicity homotopy equivalent one can construct between the two spaces to show that they are homotopic equivalent? (Which is clearly, not a strong deformation retract of any kind).